David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Synthese 170 (1):33 - 70 (2009)
David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom).
|Keywords||David Hilbert Foundations of mathematics Mathematical existence Logicism Richard Dedekind Set theory Axiomatics Mathematical logic Real number system Paradoxes Georg Cantor Consistency proofs Models|
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References found in this work BETA
William Bragg Ewald (ed.) (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Oxford University Press.
J. Ferreiros (1996). Traditional Logic and the Early History of Sets, 1854-1908. Archive for History of Exact Sciences 50:5-71.
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Jose Ferreiros (2001). The Road to Modern Logic-an Interpretation. Bulletin of Symbolic Logic 7 (4):441-484.
Friedrich Ludwig Gottlob Frege (1903). Grundgesetze Der Arithmetik Vol. (Band 2). Jena: Verlag Hermann Pohle.
Citations of this work BETA
Erich H. Reck (2013). Frege, Dedekind, and the Origins of Logicism. History and Philosophy of Logic 34 (3):242-265.
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